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On two questions concerning representations distinguished by the Galois involution

Maxim Gurevich, Jia-Jun Ma and Arnab Mitra
From the journal Forum Mathematicum

Abstract

Let E/F be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the irreducible smooth representations of GLn(E) that are distinguished by its subgroup GLn(F). One relates this class to representations which come as base change lifts from a quasi-split unitary group over F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps, we show that these two approaches are closely related. Using this observation, we are able to prove a statement relating base change and distinction for ladder representations. We then produce a wide family of examples in which the symmetry condition does not impose GLn(F)-distinction, and thus exhibit the limitations of these two approaches.

MSC 2010: 22E50; 11F70

Communicated by Freydoon Shahidi


Funding source: Israel Science Foundation

Award Identifier / Grant number: 756/12

Funding source: Hong Kong Institute of Educational Research, Chinese University of Hong Kong

Award Identifier / Grant number: CUHK 405213

Funding source: Israel Science Foundation

Award Identifier / Grant number: 1138/10

Funding statement: Maxim Gurevich, partially supported by the ISF grant 756/12, and ERC StG grant 637912. Arnab Mitra, partially supported by postdoctoral fellowships funded by the Skirball Foundation via the Center for Advanced Studies in Mathematics at Ben-Gurion University of the Negev and the Department of Mathematics, Technion.

Acknowledgements

The authors would like to thank Wee Teck Gan, Erez Lapid, Omer Offen, Dipendra Prasad, Eitan Sayag, and Jiu Kang Yu for several helpful conversations. The third author would like to thank Steven Spallone for answering his questions on parity of self-dual representations and sharing his notes on the subject matter with him. The second and the third author would like to thank the Hausdorff Institute for Mathematics (Bonn) for its warm hospitality where this project was initiated. Part of the work was done during the third author’s visit to CUHK (Hong Kong). It is a pleasure for him to thank Jiu Kang Yu for inviting him and the institute for providing an excellent work environment.

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Received: 2016-10-07
Revised: 2017-02-20
Published Online: 2017-05-05
Published in Print: 2018-01-01

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